Hessian equations with geometric applications

该项目继续了首席研究员对微分几何和非线性偏微分方程之间相互作用的研究。 特别是，它将研究某些类别的海森方程的正则性，包括四阶椭圆方程，以及欧几里得空间的体积最小化拉格朗日子流形，卡拉比-丘流形和某些伪黎曼流形。在过去的五年中，关于二阶特殊拉格朗日方程的许多问题，包括解的存在性和正则性，都取得了很大的进展。 在这个项目中，主要研究者打算研究这个方程的四阶推广，这可能与物理学更相关。 非线性四阶偏微分方程是一个年轻而令人兴奋的领域，任何进展都可能适用于其他方程，甚至其他物理科学。最近和正在进行的最优运输理论的发展表明，最优运输映射的结构与伪黎曼空间的某些极大校准子流形的几何有关。该项目的一个目标是应用校准流形的机械（如特殊拉格朗日）来获得最佳运输的新结果，从而开发一个漂亮的几何图片。该项目涉及两个令人兴奋的数学领域的研究，这似乎是联系在一起的表面。最优运输问题是如何在两个地点之间最经济有效地运输材料。这些答案直接适用于许多科学领域，包括经济学、医学成像、流体力学和气象学。发展一个强大的数学理论，强调关键要素，使那些在工业中实现基于理论的解决方案。例如，从事物流工作的人可能想知道运输某些货物的最便宜的方法。通过蛮力解决这个问题在计算上可能是不可能的，但是如果有一个好的数学理论，解决方案可以有效地计算出来。 最佳运输的另一个建议用途是创建真实的辅助外科医生的软件。 要做到这一点，一个坚实的理论是必要的。弦理论是物理学中一个令人兴奋的发展中的分支，许多人希望它能帮助我们理解宇宙的基本相互作用。在20世纪90年代后期，数学物理学家们宣称，为了更好地理解弦理论，我们应该尝试理解称为拉格朗日子流形的物体。这些对象就像具有特殊性质的极小曲面，由非线性方程控制。这个项目试图回答的问题，如当这些表面是光滑的，当他们是平坦的，当他们是不连续的。这些问题的答案将影响未来的物理学研究。 最近，首席研究员和他的合作者已经将寻找最优交通图的问题与描述某种类型的拉格朗日极小曲面的问题联系起来。极小曲面的光滑性问题已经被数学家们研究了几十年。这个项目现在想把一些几何学的思想应用到最优运输理论中。

This project continues the principal investigator's study of interactions between differential geometry and nonlinear partial differential equations. In particular, it will investigate regularity for certain classes of Hessian equations, including fourth-order elliptic equations, and also volume-minimizing Lagrangian submanifolds of Euclidean space, Calabi-Yau manifolds, and certain pseudo-Riemannian manifolds. Many questions on second-order special Lagrangian equations, including existence and regularity, have witnessed great progress over the last five years. In this project, the principal investigator intends to study the fourth-order generalization of this equation, which may be even more relevant to physics. Nonlinear fourth-order partial differential equations represent a young and exciting field, and any progress may be adaptable to other equations and even other physical sciences. Recent and ongoing developments in the theory of optimal transportation show that the structure of the optimal transportation map is related to the geometry of certain maximal calibrated submanifolds of a pseudo-Riemannian space. A goal of the project is to apply the machinery of calibrated manifolds (such as special Lagrangian) to obtain novel results in optimal transport, thereby developing a nice geometric picture.This project involves research into two exciting areas of mathematics, which appear to be linked together under the surface. The optimal transport problem asks the question of how to transport materials most cost efficiently between two locations. The answers are directly applicable in many areas of science, including economics, medical imaging, fluid mechanics, and meteorology. Development of a strong mathematical theory that emphasizes the crucial elements allows those in industry to implement solutions based on the theory. For example, someone working in logistics may want to know the cheapest way to ship certain goods. Solving the problem by brute force may not be computationally possible, but if a good mathematical theory is available, the solution can be computed efficiently. Another proposed use of optimal transportation is to create software that assists surgeons in real time. In order for this to happen, a solid theory is necessary. String theory is an exciting developing branch of physics, which many hope will lead to an understanding of the fundamental interactions of the universe. In the late 1990s, leading mathematical physicists asserted that in order to get a better understanding of string theory, we should try to understand objects called Lagrangian submanifolds. These objects are like minimal surfaces that have special properties and are governed by nonlinear equations. This project attempts to answer questions such as when these surfaces are smooth, when they are flat, and when they are discontinuous. The answers to these questions will impact the study of physics going forward. Recently, the principal investigator and his collaborators have related the problem of finding the optimal transportation map to the problem of describing a certain type of Lagrangian minimal surface. The smoothness of minimal surfaces has been intensely studied by mathematicians for decades. This project would now like to apply some of the ideas from geometry to the theory of optimal transport.